Adelic–Gamma–Theta Structure Preservation and Constant Ledgers in Spectral Approaches to the Riemann Hypothesis

We organize spectral approaches to the Riemann Hypothesis around the Adelic–Gamma–Theta factorization of the completed zeta function. The deductive spine consists of four links: a master equation independent of the zeros, a self-adjoint realization, a structural identification of the completed spectral determinant with the completed zeta function, and a structural critical-line parameterization. We prove the independence and self-adjointness links in the Rindler–Majorana, Horn, Carathéodory–Fejér, and Prime Fock frameworks. We prove the Hadamard alignment theorem, the central constant ledger, the smooth zero- counting constants for primitive Dirichlet characters, the matrix form of the Weil positivity criterion, the continuous Carathéodory–Fejér zero localization for convolution operators, a Galerkin obstruction excluding unitary equivalence between finite truncations and infinite limit operators, certified two-sided bounds for the Weil profile on [−0.8, 0.8], the Landau– Widom decay law for the Weil profile, and the interior-gap propagation theorem for finite- window Weil positivity. The global Weil positivity is reduced to the construction of outer arithmetic bounds for arbitrarily large windows. The TEBAC G1–G4 inputs, the Horn zero-set rigidity, the Yamaguchi spectral matching condition, and the Rindler–Majorana Mellin–Barnes correspondence are stated with their precise non-circularity conditions.